2010/11/12 by Armando Treibich, Treibich, Armando
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG
paper · pdf · doi:10.48550/arxiv.1011.2920
arxiv created 2010/11/12 · openalex publication_date 2010/11/12 · arxiv updated 2010/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ℙ1 and (X,q) denote, respectively, the projective line and a fixed elliptic curve marked at its origin, both defined over an algebraically closed field \mathbbK of arbitrary characteristic \emphp ≠2. We will consider all finite separable marked morphisms π:(Γ,p)→ (X,q), such that Γ is a degree-2 cover of ℙ1, ramified at the smooth point p ∈ Γ. Canonically associated to π there is the Abel (rational) embedding of Γ into its generalized Jacobian, Ap: Γ→ Jac Γ, and \0\ \subsetneq V1Γ,p...\subsetneq V gΓ,p, the flag of hyperosculating planes to Ap(Γ) at Ap(p)∈ Jac Γ (cf. 2.1. & 2.2.). On the other hand, we also have the homomorphism ιπ: X → \Jac Γ, obtained by dualizing π. There is a smallest positive integer d such that the tangent line to ιπ( X) is contained in VdΓ,p. We call it the osculating order of π. Studying, characterizing and constructing those with given osculating order d but maximal possible arithmetic genus, is one of the main issues. The other one, to which the first issue reduces, is the construction of all rational curves in a particular anticanonical rational surface associated to X (i.e.: a rational surface with an effective anticanonical divisor).